I thought, "Oh dear!" when I first saw Al Franken's public letter to Apple about the security of the new Face ID features. His question was valid, but the basic information he wanted was already on Apple's website before Franken asked about it:
Face ID is processed in the Secure Enclave and never leaves the iPhone. Nothing is sent to Apple, nothing is stored anywhere else. Everything is encrypted.
My thought is that Face Recognition--assuming it works reliably in the real world--is inherently much more secure than fingerprint detection.

However, it may turn out to seem socially a little weird to look intently at your iPhone when buying a Latte instead of discretely touching the Home button!

I think Apple actually uses the word "alert," presumably to reassure customers that no one can unlock their iPhone X by putting in front of their unconscious or drugged face. I suspect Apple could add features a user could set up, such as winking one eye, though even if that's possible, it might be a bad idea for other reasons.

"Design is not just what it looks like and feels like. Design is how it works."
Steve Jobs

In any case, I am sure that you would not be the first customer in Starbucks to be looking at an iPhone while ordering a latte!

Of course my comment was stupid. People walk around all day, in and out of traffic, in concerts, pretty much everywhere staring intently at their iPhones. I'm probably the only one who thinks that looks socially weird (not to mention dangerous). [Can't put another smilie in my reply. "The maximum number of smilies allowed is 1."]

"Design is not just what it looks like and feels like. Design is how it works."
Steve Jobs

I also get the security concerns around this. It’s not difficult to concoct some scenarios where Face ID could, in theory, be less secure than Touch ID. Someone grabs your phone from a coffee shop table and holds the phone up to you momentarily before they run off with your now-unlocked phone.

Of course, I have no inside information on Apple's R&D on face recognition, but this strikes me as exactly the kind of scenario Apple would have spent a lot of time and money on. Imagine the embarrassment (and stock-price hit) of news stories saying you just have to hold a stolen (or confiscated) iPhone X up to someone's face and it unlocks. That and an unconscious face are probably the two things Apple tested first.

Note that a photo is a nonstarter. Face ID compares a 3D wireframe model (and maybe several other factors) with the camera's view.

(Plus, in the scenario above, the legit owner could just shout "Hey Siri! Call the police; you're being stolen.")

"Design is not just what it looks like and feels like. Design is how it works."
Steve Jobs

Face ID consistently worked even without being pointed conspicuously at my face. Laying on a desk, it can generally see you from the side. No need to gaze into the device and wait for recognition. It's really fast.

In the beginning of the video, posted by Business Insider, one of the twins tries on sunglasses, a hat, a scarf, and then all three, challenging Apple's assertions that Face ID can ignore those changes. The phone passes all four tests.

When confronted with both twins sans any accessories, the phone unlocks for one of them but not the other, despite there being little difference between the two people.

One thing I haven't seen addressed is waking an iPhone X with Face ID while driving. That seems more dangerous than using Touch ID.
This also comes up if there's a passenger that wants to use the driver's iPhone X. Face ID will only accept one face, while Touch ID would accept five fingerprints, even if they're from different people. Of course, a passenger can always use the passcode.

"Design is not just what it looks like and feels like. Design is how it works."
Steve Jobs

Quite right. An excellent choice for people who get a lot of calls while driving.

I can use CarPlay (wired) or BlueTooth (which connects automatically, even if the iPhone is in my pocket) and answer the iPhone with a steering-wheel button, so it's not really an issue for me. I've done this a couple of times and it works remarkably well.
In the past, I've gotten very few calls while driving, and have no compulsion to answer immediately.

"Design is not just what it looks like and feels like. Design is how it works."
Steve Jobs